how would you solve for a limit of a sequence?

Question

blacksteel · Answer

Usually a sequence is a series of numbers found by evaluating an expression at various increasing inputs that meet some criterion. Most commonly, this is an expression in n where n is a natural number, so you are solving f(n) at n = 1,2,3,4,... to find the terms of the sequence. Then to find the limit of the sequence, you solve
$$\lim_{n \rightarrow \infty} f(n)$$
For example, suppose we are looking at the sequence generated by$$f(n) = 1/n, n \in Z^+$$ (Z^+ means the positive integers).
Then the first few terms of our sequence will be {1, 1/2, 1/3, 1/4...}
The limit of the sequence is $$\lim_{n \rightarrow \infty} 1/n = 0$$

blacksteel · Answer

I hope this answers your question, but since you asked something extremely general, post if you have any more specific questions and I'll try to answer them.

anonymous · Answer

Also, although you probably already know this, not all sequences converge.

anonymous · Answer

In other words, a limit might not exist.

anonymous · Answer

There are various ways to test for convergence but that's a bit of a digression.

anonymous · Answer

$$a _{n} := n/(n^2 +10n + 30)$$ how will I prove that the limit = 0? Thanks!

blacksteel · Answer

Good point Alch, glad you pointed that out. In general, a limit will almost always exist; what you might see is that the limit is infinity though, which means the sequence is divergent instead of convergent.

For that specific example, you can use the limit rule that in the case of a polynomial over another polynomial, the polynomial with the highest-order term will dominate. What this means is that since n^2 is of order 2 and n only of order 1, the bottom polynomial will grow faster than the top one. Thus, the ratio of the top to the bottom value will become exponentially smaller as n increases, and thus the limit of the expression is 0.

blacksteel · Answer

To show this in a more intuitive way, consider this. When n > 0, n^2 < n^2 + 10n + 30 and both expressions are positive. Thus, n^2 + 10n + 30 will grow faster than n^2. Thus, if we can prove that the limit of n/n^2 = 0 as n goes to infinity, obviously the limit of your expression must be as well since the bottom term is bigger. But n/n^2 = 1/n, whose limit is 0 as n goes to infinity.

anonymous · Answer

thank you very much. Now i understand it. so it is still somewhat the same method as solving for limits in the introduction of calculus, but now it is just applied to series and sequences? Thanks so much!